Pekka Koskela

Quasiconformal maps in metric spaces with controlled geometry

This paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry. The principal message is that such a theory is both relevant and viable. The first main issue is the problem of definition, which we next describe. Quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount. This requirement makes sense in every metric space. Given a homeomorphism f from a metric space X to a metric space Y , then for x∈X and r>0 set

Sobolev classes of Banach space-valued functions and quasiconformal mappings

We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincaré inequality.

Sobolev mappings with integrable dilatations

We show that each quasi-light mapping f in the Sobolev space W1n(Ω, Rn) satisfying ¦Df(x)¦n ≦K(x, f)J(x, f) for almost every x and for some KεLr(Ω), r>n-1, is open and discrete. The assumption that f be quasilight can be dropped if, in addition, it is required that fε W1p(ω, Rn) for some p > = n + 1/ (n-2). More generally, we consider mappings in the John Ball classes Axxxp,q(Ω), and give conditions that guarantee their discreteness and openness.

Mappings of BMO–bounded distortion

This paper can be viewed as a sequel to the work [9] where the theory of mappings of BMO–bounded distortion is developed, largely in even dimensions, using singular integral operators and recent developments in the theory of higher integrability of Jacobians in Hardy–Orlicz spaces. In this paper we continue this theme refining and extending some of our earlier work as well as obtaining results in new directions. The planar case was studied earlier by G. David [4]. In particular he obtained existence theorems, modulus of continuity estimates and bounds on area distortion for mappings of BMO–distortion (in fact, in slightly more generality). We obtain similar results in all even dimensions. One of our main new results here is the extension of the classical theorem of Painleve concerning removable singularties for bounded analytic functions to the class of mappings of BMO bounded distortion. The setting of the plane is of particular interest and somewhat more can be said here because of the existence theorem, or “the measurable Riemann mapping theorem”, which is not available in higher dimensions. We give a construction to show our results are qualitatively optimal. Another surprising fact is that there are domains which support no bounded quasiregular mappings, but admit

Definitions of quasiconformality

SummaryWe establish that the infinitesimal “H-definition” for quasiconformal mappings on Carnot groups implies global quasisymmetry, and hence the absolute continuity on almost all lines. Our method is new even inRn where we obtain that the “limsup” condition in theH-definition can be replaced by a “liminf” condition. This leads to a new removability result for (quasi)conformal mappings in Euclidean spaces. An application to parametrizations of chord-arc surfaces is also given.

Mappings of finite distortion: Capacity and modulus inequalities

Abstract We establish capacity and modulus inequalities for mappings of finite distortion under minimal regularity assumptions.

Mappings of finite distortion: Sharp Orlicz-conditions

We establish continuity, openness and discreteness, and the condition (N) for mappings of finite distortion under minimal integrability assumptions on the distortion.

Regularity of the inverse of a Sobolev homeomorphism in space

Let Ω ⊂ R be open. Given a homeomorphism f ∈ W 1,1 loc (Ω,R) of finite distortion with |Df | in the Lorentz space Ln−1,1(Ω), we show that f−1 ∈ W 1,1 loc (f(Ω),R) and that f−1 has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.

FRACTIONAL INTEGRATION, DIFFERENTIATION, AND WEIGHTED BERGMAN SPACES

We study the action of fractional differentiation and integration on weighted Bergman spaces and also the Taylor coefficients of functions in certain subclasses of these spaces. We then derive several criteria for the multipliers between such spaces, complementing and extending various recent results. Univalent Bergman functions are also considered.

Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

Abstract We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincare inequality and in addition supporting a corresponding Sobolev–Poincare-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.